Imaginary quadratic fields with class groups of 3-rank at least 2
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© Springer-Verlag GmbH Germany, part of Springer Nature 2019
Gang Yu
Imaginary quadratic fields with class groups of 3-rank at least 2 Received: 30 May 2019 / Accepted: 22 November 2019 Abstract. In this short note, we construct a family of imaginary quadratic fields whose class 1
group has 3-rank at least 2. We show that, for every large X , there are X 2 − such fields with the discriminant −D satisfying D ≤ X .
1. Introduction For any given integer g > 1, Nagell [14], Yamamoto [17] and Weigberger [16] respectively showed that there are infinitely many imaginary, and real quadratic fields with class numbers divisible by g. Quantitative results on this topic are also found in the literature, cf. [13,15], and [18]. The particular case g = 3 has received much attention. In particular, improving the results from [2,3,15], Heath-Brown [8] showed that, among the imaginary/real quadratic fields with discriminant satisfying || ≤ X , the number of those with class number divisible by 3 is 9 X 10 − . While the Cohen–Lenstra heuristics [4] suggest that a positive proportion of quadratic fields have class numbers divisible by any fixed g > 1, this exponent 9 k 10 represents a state-of-the-art lower bound for g = 2 . The search for quadratic fields with class groups having 3-rank at least 2 has also attracted quite a bit of attention. Here by 3-rank of a multiplicative Abelian group G we denote the integer r such that (Z/3Z)r G/G 3 . Let N − (32 ; X ) be the number of imaginary quadratic√fields with discriminant −D satisfying D ≤ X and such that the class group of Q( −D) has 3-rank at least 2. We are interested in getting a lower estimate for N − (32 ; X ). In Byeon [1] showed how some technical results in [18] can be used to yield a quantitative result on imaginary quadratic fields with g-rank of the class group at least 2. Byeon’s result in the case g = 3 1 gives N − (32 ; X ) X 3 − . In Luca and Pacelli [12] showed that the lower bound 1 X 3 holds for both imaginary and real quadratic fields. We shall slightly improve the exponent 13 in this note. We show G. Yu (B): Department of Mathematical Sciences, Kent State University, East Summit Street, Kent, OH 45458, USA. e-mail: [email protected] Mathematics Subject Classification: 11R11 · 11R29
https://doi.org/10.1007/s00229-019-01172-3
G. Yu
Theorem 1. For any > 0, we have 1
N − (32 ; X ) X 2 − . Notation Standard notation is used throughout. For example, τ (n) and μ(n) respectively stand for the divisor and Möbius functions; f g or f = O(g) means that there is an absolute constant c > 0 such that | f | ≤ √ cg; Cl(−D) stands for the ideal class group of an imaginary quadratic field Q( −d) with discriminant −D; by < · > and [·] we respectively denote an integral ideal with the indicated generator(s) and the ideal class that contains the indicated ideal. 2. A family of imaginary quadratic fields For positive integers n, a, and b, let f (n, a, b) := 2(a 3 + b3 )n 3 − (a − b)2 n 6 − (a 2 + ab + b2 )2 .
(2.1)
In this section, we show that, when D = f (
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